Holomorphic Dynamics on Hyperbolic Riemann Surfaces

by Marco Abate

Overview

This completely revised and updated edition of the one variable part of the author's classic older book "Iteration Theory of Holomorphic Maps on Taut Manifolds" presents the theory of holomorphic dynamical systems on hyperbolic Riemann surfaces from the very beginning of the subject up to the most recent developments. It is intended both as a reference book for the experts and as an accessible gateway to this beautiful theory for Master and Ph.D. students. It also contains extensive historical notes and references for further readings.

Product Details

ISBN-13:	9783110598742
Publisher:	De Gruyter
Publication date:	12/05/2022
Series:	De Gruyter Studies in Mathematics , #89
Sold by:	Barnes & Noble
Format:	eBook
Pages:	372
File size:	62 MB
Note:	This product may take a few minutes to download.
Age Range:	18 Years

About the Author

Marco Abate, Università di Pisa, Italy.

Table of Contents

Introduction vii

1 The Schwarz lemma and Riemann surfaces 1

1.1 The Schwarz-Pick lemma 2

1.2 The Poincare distance 8

1.3 The upper half-plane 14

1.4 Fixed points of automorphisms 19

1.5 Multipoint Schwarz-Pick lemmas 28

1.6 Riemann surfaces 36

1.7 Hyperbolic Riemann surfaces and the Montel theorem 54

1.8 Boundary behavior of the universal covering map 67

1.9 The Poincaré metric 73

1.10 The Ahlfors lemma 84

1.11 Bloch domains 88

2 Boundary Schwarz lemmas 96

2.1 The Julia lemma 97

2.2 Stolz regions and nontangential limits 113

2.3 The Julia-Wolff-Carathéodory theorem 125

2.4 The Lindelöf theorem 136

2.5 The Wolff lemma 143

2.6 The automorphism group of hyperbolic Riemann surfaces 147

2.7 The Burns-Krantz theorem 153

3 Discrete dynamics on Riemann surfaces 158

3.1 The fixed-point case 159

3.2 The Wolff-Denjoy theorem 166

3.3 The Heins theorem 169

3.4 Stability of the Wolff point 182

3.5 Models on Riemann surfaces 184

3.6 Random iteration on Bloch domains 192

3.7 Random iteration of small perturbations 201

4 Discrete dynamics on the unit disk 212

4.1 Elliptic dynamics 213

4.2 Superattracting dynamics 217

4.3 Hyperbolic dynamics 221

4.4 Parabolic dynamics 226

4.5 Models on the unit disk 234

4.6 The hyperbolic step 240

4.7 Parabolic type and boundary smoothness 251

4.8 Boundary fixed points 260

4.9 Backward dynamics 266

4.10 Commuting functions 286

5 Continuous dynamics on Riemann surfaces 294

5.1 Algebraic semigroup homomorphisms 295

5.2 One-parameter semigroups 297

5.3 One-parameter semigroups on Riemann surfaces 300

5.4 The infinitesimal generator 304

5.5 The continuous Wolff-Denjoy theorem 313

5.6 The Berkson-Porta formula 315

5.7 One-parameter semigroups on the unit disk 320

A Appendix 325

A.1 The Hurwitz theorems 325

A.2 The Fatou uniqueness theorem 326

A.3 Holomorphic functions with nonnegative real part 328

A.4 Sequences 331

A.5 Topological groups 334

Bibliography 337

Index 353